Wikipedia:Reference desk/Archives/Mathematics/2022 December 6

= December 6 =

Reason of basic cross multiplication
5 workers, need 3 h for digging a hole. If you have only 2 workers, how many hours for digging a hole of the same side would you need?

x hours / 5 workers = 3 hours / 2 workers (3/2) * 5 = 7.5

-

You need 750 gram flour for 3 pizzas, how much flour do you need for 36 pizzas?

x gr / 36 pizzas = 750 gr / 3 pizzas (36 * 750) / 3 = 9000 gr

-

I see that this works. I'm just having trouble putting the reason into words. I notice that you divide in the second case the x grams by the number of pizzas wanted, but in the first, you don't divide the x hours by the workers working x hours.

Why do we divide in the first case x hours by 5 workers? Why not divide by 2 workers? What would this mean? x hours / 2 worker = 3 hours / 5 workers Bumptump (talk) 22:14, 6 December 2022 (UTC)
 * Workers and pizzas are being treated differently because they have different roles in the problem.
 * In the first case, workers and time are both contributing (they're inputs) to make a hole (1 hole is the output). Another way to write it is (5 workers)*(3 hours) = 1 hole, so a hole requires 15 worker-hours.
 * In the second cases, flour is the only input, and pizzas are the output. 750 grams flour = 3 pizzas.--2600:4040:7B33:6E00:7083:A1B6:EE9E:2BF (talk) 22:27, 6 December 2022 (UTC)
 * It helps to make sure you've included units for all measurements. For example the first statement implies it takes (5 hours)(3 workers) = 15 worker-hours to dig the hole. The inputs are on one side of the equation and the outputs on the other side: 15 worker-hours = 1 hole. Divide the number of worker-hours by workers to get hours. In the second case, the input is flour and the output is flour, and the equation is 750 gram flour = 3 pizzas, or 250 gram flour = 1 pizza. Multiply by 36 to get the grams flour for 36 pizzas. The point is that if you keep track of what units these quantities represent, it's much easier to figure out how to combine them to make meaningful calculations. --RDBury (talk) 23:42, 6 December 2022 (UTC)
 * You can visualize it. Take the pizza example. There is this busy pizza restaurant where all pizzas are made separately to order from scratch. To accommodate the stream of orders, the kitchen has a ginormous worktop, and a large crowd of pizza workers. A separate pizza worker is assigned to each pizza that has been ordered. If 3 new orders come in, 3 pizza workers start with putting a heap of flour on the worktop. Together, these 3 heaps are 750 grams. So how much is each? H + H + H = 750 grams, so H = (750 gr) / 3 = 250 gr. Right then 33 more orders come in, so 33 more pizza workers put a heap of flour on the worktop. Now there is a line of 36 such heaps, each 250 grams. Together, H + H + H + ... + H = 36 H = 36 × (250 gr) = 9000 gr. Now replace the numbers by letters and do the algebra. To figure out the amount of resource required per item produced you use the data supplied and divide the given amount by the number of items (here 750 grams divided by 3). Then, to figure out the amount of resource needed for a requested number of items, you multiply the amount per item by that number (here 36). A sanity check is that when the number of items in the data supplied equals the requested number of items, the resulting amount of the algebra exercise should equal the original amount. (As in, "You need 750 grams of flour for 3 pizzas; how much flour do you need for 3 pizzas?") That works out: you first divide by a number and then multiply by the same number. --Lambiam 02:02, 7 December 2022 (UTC)